Boundary infinite elements for the Helmholtz equation in exterior domains

A novel approach to the development of infinite element formulations for exterior problems of time-harmonic acoustics is presented. This approach is based on a functional which provides a general framework for domain-based computation of exterior problems. Special cases include non-reflecting boundary conditions (such as the DtN method). A prominent feature of this formulation is the lack of integration over the unbounded domain, simplifying the task of discretization. The original formulation is generalized to account for derivative discontinuities across infinite element boundaries, typical of standard infinite element approximations. Continuity between finite elements and infinite elements is enforced weakly, precluding compatibility requirements. Various infinite element approximations for two-dimensional configurations with circular interfaces are presented. Implementation requirements are relatively simple. Numerical results demonstrate the good performance of this scheme. © 1998 John Wiley & Sons, Ltd.     

模拟P波向无穷远域传播的一致边界

将人工边界设置在半无穷层单元和内部有限元区域的交界面上,建立了半无穷层单元的刚度矩阵后,得到了边界节点的动力平衡方程。任意给定激励圆频率,将边界节点系统的动力平衡方程转化为特征值方程。求解特征值方程得出边界节点系统的特征值和特征模态,利用模态叠加原理得到体现左半无穷层单元和右半无穷层单元对内部有限元区域作用的边界矩阵,这就是该文的一致边界。将其与内部有限元区域的刚度矩阵进行组装来模拟无穷远域介质对波的传播作用。最后用数值算例来说明一致边界的精确性和可行性。     

High-frequency vibrations of piezoelectric plates driven by lateral electric fields

We study coupled face-shear and thickness-twist motions of piezoelectric plates of monoclinic crystals driven by lateral electric fields. The first-order theory of piezoelectric plates is used. Pure thickness modes and propagating waves in unbounded plates as well as vibrations of finite plates are studied. Both free vibrations and electrically forced vibrations are considered. Basic vibration characteristics including resonant frequencies, dispersion relations, frequency spectra and motional capacitance are obtained. Numerical results are presented for AT-cut quartz plates. The results are expected to be useful for the understanding and design of resonant piezoelectric devices using lateral field excitation.     

Two-dimensional generalized thermal shock problem of a thick piezoelectric plate of infinite extent

The theory of generalized thermoelasticity, based on the theory of Green and Lindsay with two relaxation times, is used to deal with a thermoelastic–piezoelectric coupled two-dimensional thermal shock problem of a thick piezoelectric plate of infinite extent by means of the hybrid Laplace transform-finite element method. The generalized thermoelastic–piezoelectric coupled finite element equations are formulated. By using Laplace transform the equations are solved and the solutions of the temperature, displacement and electric potential are obtained in the Laplace transform domain. Then the numerical inversion is carried out to obtain the temperature, displacement and electric potential distributions in the physical domain. The distributions are represented graphically. From the distributions, it can be found the wave type heat propagation in the piezoelectric plate. The heat wavefront moves forward with a finite speed in the piezoelectric plate with the passage of time. This indicates that the generalized heat conduction mechanism is completely different from the classic Fourier’s in essence. In generalized thermoelasticity theory heat propagates as a wave with finite velocity instead of infinite velocity in media.     

Modeling of composite piezoelectric structures with the finite volume method

Piezoelectric devices, such as piezoelectric traveling- wave rotary ultrasonic motors, have composite piezoelectric structures. A composite piezoelectric structure consists of a combination of two or more bonded materials, at least one of which is a piezoelectric transducer. Piezoelectric structures have mainly been numerically modeled using the finite element method. An alternative approach based on the finite volume method offers the following advantages: 1) the ordinary differential equations resulting from the discretization process can be interpreted directly as corresponding circuits; and 2) phenomena occurring at boundaries can be treated exactly. This paper presents a method for implementing the boundary conditions between the bonded materials in composite piezoelectric structures modeled with the finite volume method. The paper concludes with a modeling example of a unimorph structure.     

BEM for crack-hole problems in thermopiezoelectric materials

The boundary element formulation for analysing interaction between a hole and multiple cracks in piezoelectric materials is presented. Using Green's function for hole problems and variational principle, a boundary element model (BEM) for a 2-D thermopiezoelectric solid with cracks and holes has been developed and used to calculate stress intensity factors of the crack-hole problem. In BEM, the boundary condition on the hole circumference is satisfied a priori by Green's function, and is not involved in the boundary element equations. The method is applicable to multiple-crack problems in both finite and infinite solids. Numerical results for stress and electric displacement intensity factors at a particular crack tip in a crack-hole system of piezoelectric materials are presented to illustrate the application of the proposed formulation.     

Hybrid fundamental-solution-based FEM for piezoelectric materials

In this paper, a new type of hybrid finite element method (FEM), hybrid fundamental-solution-based FEM (HFS-FEM), is developed for analyzing plane piezoelectric problems by employing fundamental solutions (Green’s functions) as internal interpolation functions. A modified variational functional used in the proposed model is first constructed, and then the assumed intra-element displacement fields satisfying a priori the governing equations of the problem are constructed by using a linear combination of fundamental solutions at a number of source points located outside the element domain. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The proposed methodology is assessed by several examples with different boundary conditions and is also used to investigate the phenomenon of stress concentration in infinite piezoelectric medium containing a hole under remote loading. The numerical results show that the proposed algorithm has good performance in numerical accuracy and mesh distortion insensitivity compared with analytical solutions and those from ABAQUS. In addition, some new insights on the stress concentration have been clarified and presented in the paper.     

Electromagnetic scattering from cylindrical objects above a conductive surface using a hybrid finite-element-surface integral equation method

This work presents a novel finite-element solution to the problem of scattering from a finite and an infinite array of cylindrical objects with arbitrary shapes and materials over perfectly conducting ground planes. The formulation is based on using the surface integral equation with Green's function of the first or second kind as a boundary constraint. The solution region is divided into interior regions containing the cylindrical objects and the region exterior to all the objects. The finite-element formulation is applied inside the interior regions to derive a linear system of equations associated with nodal field values. Using two-boundary formulation, the surface integral equation is then applied at the truncation boundary as a boundary constraint to connect nodes on the boundaries to interior nodes. The technique presented here is highly efficient in terms of computing resources, versatile, and accurate in comparison with previously published methods. The near and far fields are generated for a finite and an infinite array of objects. While the surface integral equation in combination with the finite-element method was applied before to the problem of scattering from objects in free space, the application of the method to the important problem of scattering from objects above infinite flat ground planes is presented here for the first time, to our knowledge.     

Efficient numerical approach to unbounded systems subjected to a moving load

The present paper solves numerically the problem of vibrations of infinite structures under a moving load. A velocity formulation of the space–time finite element method was applied. In the case of simplex shaped space–time finite elements, the ‘steady state’ dynamic behaviour of the system was obtained. A properly performed discretization allowed of propagating information in a given direction at a limited velocity. The solutions were obtained under the assumption that the deformation is quasi-stationary, i.e., stationary in the coordinate system that moves with the load. The unbounded Timoshenko beam subjected to a distributed moving load was used as a test example. The dynamical system is placed on an elastic foundation. The matrices describing an infinite dynamical system subjected to a moving load are derived and the stability of the numerical scheme is analysed. The numerical results are compared with the analytical solutions in the literature and the classical numerical method.     

Tackling Inhomogeneous Open Boundary Designs by the Finite-Element Method

In the course of designing and analyzing electromagnetic devices, open boundary field problems often need to be solved. The mapping of the infinite exterior of a finite circular region onto the interior of that closed circular region has emerged and established itself as one of the best methods. However, as a result of the work done in the past in solving homogenous exterior regions, this paper shows that this mapping method has been mistakenly taken to apply only when the exterior consists of air or, at best, some other homogeneous medium. This paper validates the method for problems with uniformly inhomogeneous exterior regions and makes possible the solution of a new class of problems by this powerful method.     

Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis

A partial mixed finite element (FE)–state space method (SSM) semi-analytical approach is presented for the static analysis of piezoelectric smart laminate composite and functionally graded material (FGM) plates. Hence, using the Hamiltonian formalism, the three-dimensional piezoelectricity equations are first worked so that a partial mixed variational formulation, which retains the translational displacements, electric potential, transverse stresses, and transverse electric displacement as primary variables, is obtained; this allows, in particular, straightforward fulfillment of the electromechanical continuity constraints at the laminate interfaces. After an in-plane FE discretization only, the problem is first reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved using the symplectic approach; then, the multilayer solution is reached via the SSM propagator matrix. The proposed methodology is finally applied to the static analysis of piezoelectric-cross-ply hybrid laminated composite and FGM plates. In a comparison with open literature, available tabulated results show good agreements, thus validating the proposed approach.     

Free and forced vibrations of plates by boundary and interior elements

A direct boundary element method is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the static fundamental solution of the problem and this creates not only boundary integrals but surface integrals as well owing to the presence of the inertia force. Thus the discretization consists of boundary as well as interior elements. Quadratic isoparametric elements and quadratic isoparametric or constant elements are employed for the boundary and interior discretization, respectively. Both free and forced vibrations are considered. The free vibration problem is reduced to a matrix eigenvalue problem with matrix coefficients independent of frequency. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. The effect of external viscous or internal viscoelastic damping on the response is also studied. The proposed method is compared against the direct boundary element method in conjunction with the dynamic fundamental solution as well as the finite element method primarily by means of a number of numerical examples. These examples also serve to illustrate the use of the proposed method.     

A numerical solution of the steady MHD flow through infinite strips with BEM

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in infinite channels in the presence of a magnetic field is investigated. The fluid is driven either by a pressure gradient or by the currents produced by electrodes placed parallel in the middle of the walls. The applied magnetic field is perpendicular to the infinite walls which are combined from conducting and insulated parts. A boundary element method (BEM) solution has been obtained by using a fundamental solution which enables to threat the convection-diffusion type equations in coupled form with general wall conductivities. Constant elements are used for the discretization of the walls by keeping them as finite since the boundary integrals are restricted to these boundaries due to the regularity conditions as x,y→±∞. The solutions are presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number and conducting lengths. The effect of the parameters on the solution is visualized.     

Transmission line interpretation of the finite element mesh for a wave propagation problem

A one-dimensional exterior electromagnetic scattering problem is formulated using a differential equation approach followed by a finite element discretization. By interpreting the resulting linear algebraic equations as node voltage equations for a transmission line, a boundary element is obtained which satisfies the requirement of no wave reflection at the edge of the finite element region. Numerical results which show the elimination of non-physical standing waves from the scattered field are presented and discussed.     

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Recently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ??. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.     

FEM analysis of electro-mechanical coupling effect of piezoelectric materials

Based on the mechanical and electrical equilibrium equations of piezoelectric materials, the minimum potential theory is presented by using the virtual work principle in this paper. A finite element method (FEM) formulation accounting for the electro-mechanical coupling effect of piezoelectric materials is given. Some problems in the numerical simulation are discussed and the extreme illness of the stiffness matrix is overcome by the dimension changing method. As a simple application, the response of an elliptical cavity in infinite media of piezoelectric materials is analyzed. Such a geometry leads to stress and electric field concentrations.     

Vibrations of an elastic structure with shunted piezoelectric patches: efficient finite element formulation and electromechanical coupling coefficients

This article is devoted to the numerical simulation of the vibrations of an elastic mechanical structure equipped with several piezoelectric patches, with applications for the control, sensing and reduction of vibrations. At first, a finite element formulation of the coupled electromechanical problem is introduced, whose originality is that provided a set of non‐restrictive assumptions, the system's electrical state is fully described by very few global discrete unknowns: only a couple of variables per piezoelectric patches, namely (1) the electric charge contained in the electrodes and (2) the voltage between the electrodes. The main advantages are (1) since the electrical state is fully discretized at the weak formulation step, any standard (elastic only) finite element formulation can be easily modified to include the piezoelectric patches (2) realistic electrical boundary conditions such that equipotentiality on the electrodes and prescribed global charges naturally appear (3) the global charge/voltage variables are intrinsically adapted to include any external electrical circuit into the electromechanical problem and to simulate shunted piezoelectric patches. The second part of the article is devoted to the introduction of a reduced‐order model (ROM) of the problem, by means of a modal expansion. This leads to show that the classical efficient electromechanical coupling factors (EEMCF) naturally appear as the main parameters that master the electromechanical coupling in the ROM. Finally, all the above results are applied in the case of a cantilever beam whose vibrations are reduced by means of a resonant shunt. A finite element formulation of this problem is described. It enables to compute the system EEMCF as well as its frequency response, which are compared with experimental results, showing an excellent agreement. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite rotation FE-simulation and active vibration control of smart composite laminated structures

The present article focuses on the nonlinear finite element simulation and control of large amplitude vibrations of smart piezolaminated composite structures. Full geometrically nonlinear finite rotation strain–displacement relations and Reissner–Mindlin first-order shear deformation hypothesis to include the transverse shear effects are considered to derive the variational formulation. A quadratic variation of electric potential is assumed in transverse direction. An assumed natural strain method for the shear strains, an enhanced assumed strain method for the membrane strains and an enhanced assumed gradient method for the electric field is incorporated to improve the behavior of a four-node shell element. Numerical simulations presented in this article show the accurate prediction capabilities of the proposed method, especially for structures undergoing finite deformations and rotations, in comparison to the results obtained by simplified nonlinear models available in references and also with those obtained by using the C3D20RE solid element for piezoelectric layers in the Abaqus code.     

Solution of multiple crack problem in a finite plate using an alternating method based on two kinds of integral equation

This paper investigates a solution of multiple crack problem in a finite plate using an alternating method. The finite plate with cracks is an overlapping region of two regions: namely the infinite region exterior to the cracks and the finite region interior to finite plate without cracks. It is assumed that the cracks are applied by some loading and edges of the finite plate are of traction free. Governing equations for the problem and an alternating method are suggested. In the iteration, we need to solve two boundary value problems. One is the multiple crack problem in an infinite plate, and the other is the boundary value problem for the finite plate without crack. Several numerical examples are provided to prove the effectiveness of the suggested method.     

Development of semianalytical axisymmetric shell models with embedded sensors and actuators

This paper presents the development of two semianalytical axisymmetric shell finite element models, which have the possibility of having embedded and/or surface-bonded piezoelectric ring actuators and/or sensors. A mixed finite element approach is used, which combines the equivalent single-layer higher-order shear deformation theory, to represent the mechanical behavior with a layerwise discretization in the thickness direction to represent the distribution of the electrical potential of each piezoelectric layer of the frusta conical finite element. The electrical potential function is represented through a layerwise discretization in the thickness direction and can be assumed linear or quadratic with two or six electrical potential ring nodes per piezoelectric layer. The displacement field and the electrical potential are expanded by Fourier series in the circumferential direction, considering symmetric and anti-symmetric terms. Several examples are presented and discussed to illustrate the accuracy and capabilities of both models.